population growth not exponential

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  • Fri, Apr 16, 2010 - 12:07am

    #21
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    Re: population growth not exponential

Piquod12 seems to be very lonely here, with everyone telling him he is wrong.

I believe that many are reading his posts wrong, since I believe his points are perfectly true.

What he is saying is that the growth rate is dropping.
In fact if you look over the last 20 years at the birthrates it is dropping in almost every country of the world.
The important turningpoint is when the worlds fertillity rate is at or below 2.1 childs per woman, as that is the rate for constant population in the long term.

Piquod12 also mentions (but not in these excat words) that average lifespan is increasing.
If we for the sake of simplicity say that the average lifespan will increase to a certain level and then stop, then the increased lifespan will increase world population by a factor k. Lets NOT go into what exactly this factor is. The important point is that it is a constant, and not anything that leads to exponential growth.
This is assuming you believe there is a maximum age that humas can become. If you believe there will ever be a way for eternal life for humans then we are in deep trouble again.

So, even if population are still growing for the next 40 years it is a big difference if we will be at a point when population will start to decline at 2050 or if population is still growing a lot.

Just my 2 cents to show my support for Piquod12 for bringing up this subject.

  • Fri, Apr 16, 2010 - 10:02am

    #22
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    Re: population growth not exponential

Hi Mike

Sorry mate, but YOU are plain wrong……  you can only get a plateau at 0%.

All that happens when you lower the number in front of the % sign is you change the shape of the curve.  Yes it’ll take longer (maybe a lot longer!), but eventually the curve will again go skywards, ESPECIALLY as it starts from an already very large number, unlike the current one which starts (pick a date, any date) around 2000 years ago when world population was 200 million as opposed to today’s 7,000 million….

It’s possible we’re arguing at cross-purposes here because I don’t disagree with any of the above. What I would say is that the type of growth you describe is not exponential. This really isn’t a matter for debate but simply a consequence of a mathematical definition that describes the rate of growth of a function as proportional to the function’s current value and where that rate is a constant;

A quantity x depends exponentially on time t if

x(t)=acdot b^{t/tau},

where the constant a is the initial value of x,

x(0)=a, ,

and the constant b is a positive growth factor, and τ is the time required for x to increase by a factor of b:

x(t+tau)=x(t)cdot b, .

My bold.

So whilst it is perfectly valid to say population is still growing it is not valid to describe the growth as exponential*. You can say this is nitpicking but population projections based on continued exponential growth would be very different from those based on current growth trends (i.e. plateauing around 9bn).

*Sofisteks example of exponential growth at one rate followed by exponential growth at a diferrent rate, apart from being highly unlikely, is also not an example of overall exponential growth. I’m not sure what it is an example of, if it even has a name, but you can’t just stitch curves of different shape together and give the result the same label as the components.

I should also make clear (in case I’ve created the wrong impression) that I do not believe 9bn to be a sustainable level for our planet, especially once economic growth projections are factored in! I would also find it hard to argue with anyone claiming the current population X gdp product is sustainable. But until this has been demonstrated by the passage of time it remains in the realm of opinion. The mathematical definition of exponential growth is not.

  • Fri, Apr 16, 2010 - 10:44am

    #23
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    Re: population growth not exponential

Hi Silvervarg

Support appreciated thanks 🙂

To be honest I’m not that surprised I’ve ruffled a few feathers on here by introducing myself with a criticism of part of CM’s presentation. With hindsight it probably wasn’t the most tactful approach! It’s just that I believe that fundamental assumptions (whether intended or not) need to be regularly reviewed and challenged in order to make sure we don’t get into group-think which ends up helping no-one. It’s just too easy to get sucked into a paradigm, especially if there is an inclination already present, and filtering all new information through the same lens.

Which is why in a seperate thread I’ve challenged the perception of our dependence on oil by questioning how dependent we are on the biggest user – the private car. Of course that is a discussion which is much more in the realm of opinion than fact and has massively different implications depending on where in the World you live. E.g. people in the US may see private car loss it as an insurmountable issue whilst those in Europe might view it as just an inconvenience.

All worthy of discussion IMO and if it makes me unpopular then so be it. Since when have scientists been popular anyway, look at what they did to Galileo 😉

  • Fri, Apr 16, 2010 - 08:40pm

    #24
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    Re: population growth not exponential

[quote=piquod12]Sofisteks example of exponential growth at one rate followed by exponential growth at a diferrent rate, apart from being highly unlikely, is also not an example of overall exponential growth. I’m not sure what it is an example of, if it even has a name, but you can’t just stitch curves of different shape together and give the result the same label as the components.[/quote]Of course you can – that is what just about every growth graph I’ve ever seen does. Your idealised notion of exponential growth is not, in the least, important, from a real world perspective. My example of exponential growth at one rate, followed by exponential growth at a different rate is certainly not “highly unlikely”; it is a fact. Even your idealised exponential curves are a specific example (where X=Y, when the growth rate of one period is X and the growth rate of the next period is Y). All you have to do is keep shrinking the time period to get to a period where the growth rate is constant for that period. The subsequent time period can also be shrunk, until the same condition is met. Concatenate all of these time periods and, provided each period has growth, you have exponential growth throughout the whole period – just not the idealised version that you think is important.

 

The critical feature is that any percentage growth is on an increasing population size.

It’s not a matter of rustling feathers, it’s just that your criticism has no merit, regardless of how mathematically accurate it may be.

  • Sat, Apr 17, 2010 - 07:04am

    #25
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    Re: population growth not exponential

Hi Sofistek,

My example of exponential growth at one rate, followed by exponential growth at a different rate is certainly not “highly unlikely”; it is a fact

Can you give a real World example where this has occurred? I don’t mean a continuously varying growth rate but  where growth at one rate constant is replaced by growth at a different rate constant, i.e. two actual exponential in sequence.

Of course idealised exponential growth will not occur in nature either, there will always be some variation in the rate. But it is still valid to talk about the importance of exponential growth even if it only an approximation and to be clear to distinguish between systems that exhibit it and those that do not. The merit of this distinction comes in the nature of reasonable projections going forward. As I’ve said, the difference between an exponential projection and that obtained from a declining rate is not trivial. It is extremely pertinent to what is most likely to happen to population in the future and how we go about dealing with it.

You talk about shrinking the time interval to a point of constant growth rate, essentially taking the derivative of the curve. Yes you can do that and, yes, you can express the level of growth as a function of the population size but that doesn’t make the curve exponential. Even linear growth can, at any infinitessimally small time interval, be exressed as a % of the underlying but it doesn’t mean the linear growth is now exponential! Take a simple example where, say, the underlying population is 6bn and it’s increasing linearly by 70mn/year (remember this is an example not my belief about what is actually happening). At the point at which the population is 6bn then you can express the growth as either 70mn/year or 70/6000 X 100% = 1.17%. The following year 70mn peolple will have been added and you can, at the specific point of year-end, once again express growth as a %, this time it will be 70/6070 X 100% = 1.15%. The numerator (top bit of the equation) is still 70 because we have defined linear grwoth. But by your argument you have exponential growth because growth can still be expressed as a % of the underlying population. It isn’t, it’s linear.

The critical feature is that any percentage growth is on an increasing population size

Critical in what way? If the rate is falling quickly enough then the absolute growth, exressed as a number, will also be falling regardless of the increasing size of the function – i.e. you will get plateauing rather than continued acceleration.

  • Sat, Apr 17, 2010 - 09:25pm

    #26
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    Re: population growth not exponential

We seem to be on a completely different wavelength, here, piquod12.

By your reckoning, there is no real world example of exponential growth, because, taken over the whole time period of that growth there cannot be a constant rate of growth and, so, no exponential growth. If you deny this for, say, population during some period then you have to limit the time period to get that roughly constant rate. All I’m saying is that all examples of growth can be seen to be exponential over a short enough time period (which you acknowledge) so that, when talking about that growth at any particular instant in time, it will always be exhibiting exponential growth, even though you’re right that it won’t be a pure exponential graph over the whole time frame of that growth.

Your notion of linear growth can be as easily criticised as Chris’s hockey stick because it would imply that the linear rate would continue ad infinitum, which no-one could possibly know (aside from the near certainty that the increase would vary and so not be linear). However, even linear growth would have horrendous implications somewhere down the line, since some quantity of humans (probably in the tens of millions) would be added to the planet every year. I prefer to call it exponential growth because it can more clearly demonstrate the enormity of our predicament.

This is why your, possibly valid, mathematical criticism of that one part of the Crash Course, is irrelevant.

  • Sun, Apr 18, 2010 - 06:38am

    #27
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    Re: population growth not exponential

Hi Sofistek

By your reckoning, there is no real world example of exponential growth, because, taken over the whole time period of that growth there cannot be a constant rate of growth and, so, no exponential growth.

No, I’m happy to acknowledge exponential growth (or approximation to the same) for systems that exhibit it for a relevent period of time. By relevent I mean finite (as opposed to instantaneous) and of sufficient duration to be meaningful from the pov of the system in question. I absolutely agree that human population growth has, in the past, exhibited exponential growth.

when talking about that growth at any particular instant in time, it will always be exhibiting exponential growth,

No this bit is technically wrong – you cannot actually determine the shape of a growth curve by by taking the derivative at one time point. Hence my example of linear vs exponential growth. One simply doesn’t have enough information.

However, even linear growth would have horrendous implications somewhere down the line, since some quantity of humans (probably in the tens of millions) would be added to the planet every year. I prefer to call it exponential growth because it can more clearly demonstrate the enormity of our predicament.

I agree that linear growth would also eventually hit limits on a finite planet. It would take longer but yes it would get there in the end. If you like to call linear growth exponential then I guess that’s your call and I’m getting to the point where I no longer believe I can persuade you otherwise.

This is why your, possibly valid, mathematical criticism of that one part of the Crash Course, is irrelevant.

I maintain that the shape of the population curve going forward is relevent. For one thing the amount of time we will have to adapt will be very dependent on the speed with which things are likely to change. Same for oil depletion – if it happens at 10%/year it will have very different consequences than at 2%/year. I’m not arguing against the seriousness of our predicament but do believe the details matter.

Anyway I’m aware that I’m starting to repeat myself here and therefore not really adding anything new. So, unless anyone has a specific question for me, I’ll stop for now.

  • Sun, Apr 18, 2010 - 08:09pm

    #28
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    Re: population growth not exponential

piquod12,

I get your point and, in pure mathematical terms, you have a valid point but has little value in reality unless you can show that the future will pan out in a way that has population levelling out at some figure (say, 10 billion), because, if you could do that, we’d at least have a maximum figure to deal with. However, if we’re already in overshoot, it seems to me that it makes little difference whether some idealised curve will see a maximum number in 2050 or 2100 or 2200. Chris has focused on what the historical trend was and we need to pay attention to that. Forecasts for future trends are liable to be wrong (almost all predictions about the future are destined to be wrong) and population growth didn’t stick to the projected trend between 2004 and 2009, so who knows what the future holds in terms of population size?

We need hard hitting messages to try to get a significant shift in thinking and to get a new strategy for a sustainable future. It seems to me that a mathematical criticism that some strict definition of a word means it is the wrong word to lose, is largely irrelevant. There is a grave story to tell and it needs to be told, without picky side threads that could dilute the message.

By the way, you’re right about my use of the word instantaneous but the fact that you picked up on it shows that you tend to miss the primary message, which is what you did with this thread.

  • Wed, Apr 28, 2010 - 07:50am

    #29
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    Re: population growth not exponential

you’re right it’s not exponential – it’s worse

Mark

  • Sun, Jan 23, 2011 - 01:45am

    #30
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    A common problem with real life data

In enjoyed the discussion here as there were interesting arguments from various sides which do inspire thinking..

As a physicist I tend to agree with piquod12 and Chris who could find peace with each other. Piquod12 s observation of a recent change of the rate of growth was something I also noticed with some interest when I check the numbers at Wikipedia after watching the crash course. Piqued12, I believe just wanted to help Chris fix an inaccuracy to make him less vulnerable to more venomous attacks.

I believe piquod12 is right with the mathematics and of course a curve with changing exponential growth rates cannot be an overall exponential curve at the same time.

Maybe this is more obvious to the non mathematician like myself if you think of a curve that changes its linear growth rate i.e. its “steepness”.

Is the result of a linear function that changes rates (even if limited to positive numbers) linear?
No. Nobody would call such a curve as overall linear as it could describe any zigzag patterns or castle patterns or even exponential patterns. In fact, the only thing it cannot describe is …. linear behaviour, a straight line.

Also: Even though a changing rate “linear” function can also describe an exponential function, it would not make sense to claim an exponential function is linear.

Even simpler: it is not really useful to consider a curve to be straight even though one can describe a curve as a straight line with a varying degree of “bendness”…

A varying rate of an exponential curve will also be a lot of different things but certainly not overall exponential.

However, these are academic arguments. Mathematicians are the only people I know who can claim truth and perfection – as long as you can’t question the accuracy and meaning of the data they play with.
Interestingly the moment we leave the mathematical truth behind, we can call it science Wink

Scientist like anyone else have to live with unavoidable errors in our data.

Nobody here talked about the error bar in the population data. Do we really know how many Africans or Indians or Chinese people there are?

Many people in these countries do not have birth certificates or pay rent or taxes or even official names… and even if we could determine their number counting income tax returns, do all countries count their tax returns (or census forms) at the same time?

We also don t know the global population 200 years ago – the error for the data in Chris graph for that time is likely considerable. The data he got were probably already extrapolated assuming exponential growth until more reliable global data became available.

The graph in the previous post is a reminder that it is scientifically and mathematically incorrect (although not uncommon depending on the type of publication) to connect uncertain data points and trying to call the resulting shape something e.g. exponential.

The scientifically correct procedure would be (in simple terms) to fit the data points including their error bars (if known) with a mathematical function that would make sense while trying to keep the deviation from the data points a minimum.

Assuming reasonable error bars of the data, nobody would seriously suggest to use a linear function if we look at , say a few thousand years of human population growth. It is most likely an exponential curve that changes its rate as human fertility and mating behaviour may depend on changing external  factors, climatic changes, wars, cultural changes over time.

However, since the raw data we have are probably rather inaccurate, science tells us to fit a simple exponential function until the very last data points and further..

Note that depending on the errors and the unknown true growth function the fitted exponential function would not go through the data points but generally only approximate them.

Now we are back to the beginning of the discussion of this interesting (for some) thread. Chris’ representation makes practical sense – but he could have saved himself and us some time arguing by adding error bars to his data points or at least mention their uncertainty.

Now, before someone comments that data in the last years are probably more accurate and may justify a linear fit (which is not shown in the previous posts graph due to the large x scale): There may also be higher incentives to manipulate these data (I would expect a larger and perhaps asymmetric error bar).

In a time dominated by a profit and growth driven economy that benefits a few percent of the population – can we trust the same few percent to give us accurate numbers supporting exponential growth – a threat to their “values” and financial position?

Let us rather be careful..and focus on the larger picture and common sense which is more reliable nowadays – like Chris suggests.

 

 

 

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